categorical direct product is an inverse limitCategoricalDirectProductIsAnInverseLimit2004-02-25 01:21:422004-02-25 01:21:42Theoremlimit of a functor% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\Hom}{Hom}\begin{theorem}
The categorical direct product can be realized as an example of an inverse limit.
\end{theorem}
\begin{proof}
Suppose we have a direct product of $\{C_i\}_{i\in I}$ for some ($\Univ$) set $I$.
Consider $I$ as a category whose arrows are only the identity arrows. Then we can
define a functor $G$ by $G(i)=C_i$. It is then clear that the universal property
of an inverse limit is equivalent to the universal property defining a categorical
direct product.
\end{proof}
Reversing the arrows, it is also clear that the categorical direct sum is an example of a direct limit.
These results are of interest when one is looking to prove exactness of sums and products in a category: often it is easier to address exactness of direct and inverse limits, and the result then applies to many other constructions as well.